Scalar Curvature Question [?27]: On the other hand, if theδ-neighbourhoods Uδ(S) ⊂X of allT(X)- non-spin surfacesS in a Riemannian manifoldX ar

Definitions and mathematical statement

Let $(X,g)$ be a Riemannian manifold and let $\delta>0$. For a surface $S$ in $X$, its metric $\delta$-neighborhood is \[ U_\delta(S)=\{x\in X:\operatorname{dist}_g(x,S)<\delta\}. \] The source singles out "$T(X)$-non-spin surfaces." This refers to the spin behavior of the ambient tangent bundle along a surface, not automatically to the surface's own tangent bundle. In an oriented pullback-bundle interpretation, one would test the second Stiefel--Whitney class of the restricted or pulled-back $TX$, but the source gives no exact surface or bundle convention.

The proposed qualitative implication is that if the neighborhoods $U_\delta(S)$ of every such surface are "large," then the spin area of $X$ must also be large. The supplied record defines neither spin area nor largeness. In particular it does not specify whether neighborhood size means volume, diameter, width, a surface-filling quantity, or another invariant; nor does it prescribe any lower bound or dependence on $\delta$.

A complete quantitative statement would require an admissible surface class, the neighborhood-size functional, a precise spin-area invariant, and the inequality linking their values. It must also address the case of no qualifying non-spin surfaces and specify the dimension, compactness, and orientation assumptions on $X$. These choices cannot be recovered from the qualitative sentence and are not silently supplied.

Short English statement

Does uniformly large neighborhood geometry around all ambient-tangent-non-spin surfaces force large spin area of the manifold? The relevant invariants and thresholds are missing, so the source gives a speculative program rather than a quantified inequality.

The supplied summary connects the idea with spin area, $K$-area, slice-area, and two-dimensional scalar-curvature obstructions, but says no direct theorem verifying the stated implication was found. It gives no definition equating those notions. The OPEN-TRIAGE label and 2017 passage are retained with current status needing review.

Sources

• Local UnsolvedMath registry, permanent ID 6700027, source code catalogue entry, scalar-curvature Question [?27], proposed by Misha Gromov, 2017. Dataset curation and metadata: Ulam AI and UnsolvedMath Contributors, CC BY 4.0.
• Misha Gromov, 101 Questions, Problems and Conjectures around Scalar Curvature (2017), printed tag [?27], occurrence 28, PDF page 38, https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf; normalized pdf-text extraction, accessed 2026-07-29.
• No exact later theorem or independent definition of spin area is supplied. Source-specific provenance: the author-hosted PDF is incomplete and publicly available; current status and statement-reuse terms NEEDS_REVIEW, separately from the dataset licence.